where [Greek: theta] is the angle the radius vector makes with the direction of motion. The lines of magnetic force are circles around the axis of motion. When the ion is moving with a velocity small compared with the velocity of light, the lines of electric force are nearly radial, but as the speed of light is approached, they tend to leave the axis of motion and to bend towards the equator. When the speed of the body is very close to that of light, the magnetic and electric field is concentrated to a large extent in the equatorial plane.
The presence of a magnetic field around the moving body implies that magnetic energy is stored up in the medium surrounding it. The amount of this energy can be calculated very simply for slow speeds.
In a magnetic field of strength H, the magnetic energy stored up in unit volume of the medium of unit permeability is given by H^2/(8π). Integrating the value of this expression over the region exterior to a sphere of radius a, the total magnetic energy due to the motion of the charged body is given by
[integral]_{a}^[infinity] (H^2/(8π))d(vol) = (e^2u^2/(8π))[integral]_{0}^2π [integral]_{0}^π [integral]_{a}^[infinity] (sin^2 θ)/(r^4)r(sin θ)dφrdθdr
= (e^2u^2)/4[integral]_{0}^π [integral]_{a}^[infinity] ((1 - cos^2 θ)/r^2)(sin θ)dθ.dr
= (e^2 u^2)/3 [integral]_{a}^[infinity] dr/r^2 = e^2 u^2/(3a).
The magnetic energy, due to the motion, is analogous to kinetic energy, for it depends upon the square of the velocity of the body. In consequence of the charge carried by the ion, additional kinetic energy is associated with it. If the velocity of the ion is changed, electric and magnetic forces are set up tending to stop the change of motion, and more work is done during the change than if the ion were uncharged. The ordinary kinetic energy of the body is (1/2)mu^2. In consequence of its charge, the kinetic energy associated with it is increased by (e^2 u^2)/(3a). It thus behaves as if it possessed a mass m + m_{1} where m_{1} is the electrical mass, with the value 2e^2/(3a).
